Chapter 14, Problem 87P

(a)

To determine

The amplitude of oscillation.

Explanation of Solution

Given:

Mass of object is $2.00\text{kg}$ .

Linear damping constant is $2.00\text{kg}/\text{s}$ .

Value of force constant is $400\text{N}/\text{m}$ .

Value of maximum force is $10.0\text{N}$ .

Angular frequency is $10.0\text{rad}/\text{s}$ .

Formula used:

Write expression for amplitude of damped oscillation.

  $A=\frac{F_{\max }}{\sqrt{m^2\left({\omega }_0^2-{\omega }^2\right)+b^2{\omega }^2}}$

Here, $F_{\max }$ is the maximum force, $m$ is the mass of object, $b$ is the damping constant and $\omega$ is the angular frequency.

Substitute $\sqrt{\frac{k}{m}}$ for ${\omega }_0$ in above expression.

  $A=\frac{F_{\max }}{\sqrt{m^2\left(\frac{k}{m}-{\omega }^2\right)+b^2{\omega }^2}}$   …… (1)

Calculation:

Substitute $10\text{N}$ for $F_{\max }$ , $2\text{kg}$ for $m$ , $400\text{N}/\text{m}$ for $k$ , $10.0\text{rad}/\text{s}$ for $\omega$ and $2.00\text{kg}/\text{s}$ for $b$ in equation (1).

  $\begin{array}{c}A=\frac{10\text{N}}{\sqrt{{\left(2\text{kg}\right)}^2\left(\frac{400\text{N}/\text{m}}{2\text{kg}}-{\left(10.0\text{rad}/\text{s}\right)}^2\right)+{\left(2.00\text{kg}/\text{s}\right)}^2{\left(10.0\text{rad}/\text{s}\right)}^2}}\\ =0.0498\text{m}\end{array}$

Conclusion:

The amplitude of oscillation is $0.0498\text{m}$ .

(b)

To determine

The frequency at which resonance occurs.

Explanation of Solution

Given:

Mass of object is $2.00\text{kg}$ .

Linear damping constant is $2.00\text{kg}/\text{s}$ .

Value of force constant is $400\text{N}/\text{m}$ .

Value of maximum force is $10.0\text{N}$ .

Angular frequency is $10.0\text{rad}/\text{s}$ .

Formula used:

Write expression for condition of resonance.

  $\omega ={\omega }_0$

Substitute $\sqrt{\frac{k}{m}}$ for ${\omega }_0$ in above expression.

  $\omega =\sqrt{\frac{k}{m}}$   …… (2)

Calculation:

Substitute $2\text{kg}$ for $m$ and $400\text{N}/\text{m}$ for $k$ in equation (2).

  $\begin{array}{c}\omega =\sqrt{\frac{400\text{N}/\text{m}}{2\text{kg}}}\\ =14.14\text{rad}/\text{s}\end{array}$

Conclusion:

Thus, the frequency at which resonance occurs is $14.14\text{rad}/\text{s}$ .

(c)

To determine

The amplitude of oscillation at resonance.

Explanation of Solution

Given:

Mass of object is $2.00\text{kg}$ .

Linear damping constant is $2.00\text{kg}/\text{s}$ .

Value of force constant is $400\text{N}/\text{m}$ .

Value of maximum force is $10.0\text{N}$ .

Angular frequency is $10.0\text{rad}/\text{s}$ .

Formula used:

Write expression for amplitude of damped oscillation.

  $A=\frac{F_{\max }}{\sqrt{m^2\left({\omega }_0^2-{\omega }^2\right)+b^2{\omega }^2}}$   …… (3)

Here, $F_{\max }$ is the maximum force, $m$ is the mass of object, $b$ is the damping constant and $\omega$ is the angular frequency.

Write expression for condition of resonance.

  $\omega ={\omega }_0$

Substitute $\omega$ for ${\omega }_0$ in equation (3)

  $A=\frac{F_{\max }}{\sqrt{m^2\left({\omega }^2-{\omega }^2\right)+b^2{\omega }^2}}$

Solve above expression.

  $A=\frac{F_{\max }}{\sqrt{b^2{\omega }^2}}$   …… (4)

Calculation:

Substitute $10\text{N}$ for $F_{\max }$ , $10.0\text{rad}/\text{s}$ for $\omega$ and $2.00\text{kg}/\text{s}$ for $b$ in equation (4).

  $\begin{array}{c}A=\frac{10\text{N}}{\sqrt{{\left(2.00\text{kg}/\text{s}\right)}^2{\left(10.0\text{rad}/\text{s}\right)}^2}}\\ =0.0354\text{m}\end{array}$

Conclusion:

Thus, the amplitude of oscillation at resonance is $0.0354\text{m}$ .

(d)

To determine

The width of resonance curve.

Explanation of Solution

Given:

Mass of object is $2.00\text{kg}$ .

Linear damping constant is $2.00\text{kg}/\text{s}$ .

Value of force constant is $400\text{N}/\text{m}$ .

Value of maximum force is $10.0\text{N}$ .

Angular frequency is $10.0\text{rad}/\text{s}$ .

Formula used:

Write expression for width of resonance curve.

  $\Delta \omega =\frac{b}{m}$   …… (5)

Here, $\Delta \omega$ is the width of resonance curve.

Calculation:

Substitute $2\text{kg}$ for $m$ and $2.00\text{kg}/\text{s}$ for $b$ in equation (5).

  $\begin{array}{c}\Delta \omega =\frac{2.00\text{kg}/\text{s}}{2\text{kg}}\\ =1\text{rad}/\text{s}\end{array}$

Conclusion:

Thus, the width of the resonance curve is $1\text{rad}/\text{s}$ .